

Algorithm: Grid-based conditional sampling

Fix the boundary probabilities \(p_{\min },p_{\max }\in (0,1)\)  and compute the boundary quantiles \(S_{\min }:=F_{S(T)}^{-1}(p_{\min })\)  and \(S_{\max }:=F_{S(T)}^{-1}(p_{\max })\)  (e.g. with the COS method [1]}).

Compute the reference values \(S^q:=S_{\min }+\frac{q-1}{Q-1}(S_{\max }-S_{\min })\) , \(q=1,\dots ,Q\) .

Given the “augmented” parameters \(\mathbf {p}^q\) , evaluate \(Q\)  times \(\tilde{H}\)  to compute \(\mathbf {G}\)  (see (REF )).

Given the parameters \(\mathbf {p}\)  and the grid \(\mathbf {G}\) , for \(j=1,\dots ,N_{\texttt {paths}}\) , repeat:

generate the sample \(\hat{S}_j\)  from \(S(T)\)  (e.g. with COS-SC method [1]} and [3]});

given \(\hat{S}_j\) , approximate the \(M\)  conditional CVs, i.e. \(\mathbf {a}_{\hat{S}_j}\) , by interpolation in \(\mathbf {G}\) ;

given \(\mathbf {a}_{\hat{S}_j}\) , use SC to generate the conditional sample \(\hat{A}_j\) .





